Applied Time Series Notes ( 1)

Transcription

1 Applied Time Series Notes ( 1) Analytics - Time Series I Preliminaries: Dates ñ Internal: # days since Jan 1, 1960 ñ Need format for reading, one for writing ñ Often DATE is ID variable (extrapolates) ñ Program has lots of examples: options ls=76 nodate; title "Time Series Example 1"; data A; input date $ cards; Jan82 10 Apr82 30 Jul82 60 Oct82 40 Jan83 50 Feb83 20 Mar83 35 ; proc plot; plot Y*date/vpos=20 hpos=50;run; data B; Diff = '01jan80'D-'13nov1979'd; dec1919='31dec1919'd; dec19='31dec19'd; jan20='01jan20'd; date1 = 18; date2=18; date3=18; format date1 monyy.; format date2 date9.; format date3 mmddyy.; proc print; run;

2 Applied Time Series Notes ( 2) data c; array date(3); input x date7. ; do i=1 to 3; twoi=2*i; date(i) = intnx('month','01jan1912'd,twoi); end; cards; 01feb60 ; proc print; proc print; format date1-date3 mmddyy.; data next; set a; newdate= input(date, monyy.); newdate2 = newdate; format newdate2 monyy.; proc sort; by date; proc print; proc sort; by newdate2; proc print; run; Program Output: Time Series Example 1 Plot of Y*date. Legend: A = 1 obs, B = 2 obs, etc. Y 60 ˆ A 55 ˆ 50 ˆ A 45 ˆ 40 ˆ A 35 ˆ A 30 ˆ A 25 ˆ 20 ˆ A 15 ˆ 10 ˆ A Šƒˆƒƒƒƒƒƒˆƒƒƒƒƒƒˆƒƒƒƒƒƒˆƒƒƒƒƒƒˆƒƒƒƒƒƒˆƒƒƒƒƒˆƒƒƒ Apr82 Feb83 Jan82 Jan83 Jul82 Mar83 Oct82 date

3 Applied Time Series Notes ( 3) Time Series Example 1 Obs Diff dec1919 dec19 jan20 date1 date2 date JAN60 19JAN /19/60 Obs date1 date2 date3 x i twoi Obs date1 date2 date3 x i twoi 1 03/01/12 05/01/12 07/01/ Obs date Y newdate newdate2 1 Apr APR82 2 Feb FEB83 3 Jan JAN82 4 Jan JAN83 5 Jul JUL82 6 Mar MAR83 7 Oct OCT82 Obs date Y newdate newdate2 1 Jan JAN82 2 Apr APR82 3 Jul JUL82 4 Oct OCT82 5 Jan JAN83 6 Feb FEB83 7 Mar MAR83

4 Applied Time Series Notes ( 4) PROC EXPAND (cubic spline) data last; input date=intnx('month','01dec83'd,_n_); format date monyy.; cards; ; proc print; proc expand data=last from=month to=month out=out1; convert Y = ynew; id date; data out1; merge out1 last; by date; proc plot data=out1; plot Y*date="*" Ynew*date = "-" /overlay vpos=20 hpos=50; proc expand data=last from=month to=week out=out2 outest=spline; convert Y = Ywk; id date; proc print data=out2(obs=5); proc print data=spline(obs=5); data out2; merge out2 last; by date; proc plot data=out2; plot Y*date="*" Ywk*date = "-" /overlay vpos=20 hpos=50; run;

5 Applied Time Series Notes ( 5) Time Series Example 1 Obs Y date 1 10 JAN84 2. FEB84 3. MAR APR MAY JUN84 7. JUL AUG SEP OCT NOV DEC JAN FEB MAR APR85

6 Applied Time Series Notes ( 6) Plot of Y*date. Symbol used is '*'. Plot of ynew*date. Symbol used is '-'. Y 40 ˆ * ˆ - - * * * - * * * * * - 0 ˆ Šˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒ JUL83 FEB84 AUG84 MAR85 date NOTE: 7 obs had missing values. 9 obs hidden. Obs date Ywk 1 Sun, 1 Jan Sun, 8 Jan Sun, 15 Jan Sun, 22 Jan Sun, 29 Jan

7 Applied Time Series Notes ( 7) date CONSTANT LINEAR QUAD CUBIC DEC JAN MAY JUN AUG Time Series Example 1 Plot of Y*date. Symbol used is '*'. Plot of Ywk*date. Symbol used is '-'. Y 40 ˆ * ˆ * - * *-- -- * *- -* *--* ˆ Šˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒ Sun, 1 Jan 1984 Sun, 2 Dec 1984 Sun, 3 Nov 1985 date NOTE: 83 obs had missing values. 17 obs hidden.

8 Applied Time Series Notes ( 8) Data Sets ñ Expect column for id (date) ñ Dependent (response, target) variable and explanatory (input) variable columns ñ May need to transpose, combine. data a; input y1-y5; cards; data b; retain date; if _n_=1 then do; input month day date=mdy(month,day,year); end; input Y if _n_>1 then date=date+1; format date mmddyy.; cards; ; proc print data=a; proc print data=b; proc transpose data=a out=aa; var y1-y5; data aa; set aa (rename=(col1=y)); *drop _name_; date=date+1; retain date '29oct1987'd; proc print data=aa; data both; merge b aa; by date; proc print; run; Obs y1 y2 y3 y4 y

9 Applied Time Series Notes ( 9) Obs date month day year Y Z 1 10/28/ /29/ /30/ /31/ /01/ /02/ /03/ /04/ Obs _NAME_ Y date 1 y y y y y Obs date month day year Y Z _NAME_ 1 10/28/ /29/ /30/ y1 4 10/31/ y2 5 11/01/ y3 6 11/02/ y4 7 11/03/ y5 8 11/04/

10 Applied Time Series Notes ( 10) ñ Merge: Format is first one encountered (aa has none) ñ Merge: Value is last one encountered (like overrecording a tape) Transformations ñ Most common: no transformation or log Try log if variation increases as mean increases. ñ Log difference often used in economics (Log is natural log, ln) Log(Y t) - Log(Y t-1) = Log(Y t/y t-1) 2 Taylor's series: Log (1+%) = log(1) + 1 % - % /2+ â % for % small. Thus 100 Log(Y t/y t-1) is approximate percentage change if small. data a; array X(11); do i=1 to 11; X(i) =.88+i/50; end; output; do i=1 to 11; x(i) =log(x(i)); end; output; proc print noobs; var X1-X11; format X1-X11 5.2; run; X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X

11 Applied Time Series Notes ( 11) ñ log(y)=x µ N(, ) => E{Y} = E{e } = " # X ÐX-. Ñ2 ' È e e 25 2 dx = 15 " # X 2-2( )X +. 2 ' È e 25 2 dx = 15 " È# X 2 X 2-2( )X + ( ) /2 e ' e 25 2 dx = /2. e (not just e ) ñ Although exponentiating mean of log(y) does not give C mean of Y, it is true that Pr{log(Y) <C} = Pr{Y<e } from which we see that exponentiating median of log(y) gives. the median of log(y), that is, e is median of Y (not mean).